Question: You have found the following ages (in years) of all 6 sloths at your local zoo: $ 11,\enspace 3,\enspace 9,\enspace 8,\enspace 2,\enspace 15$ What is the average age of the sloths at your zoo? What is the variance? You may round your answers to the nearest tenth.
Answer: Because we have data for all 6 sloths at the zoo, we are able to calculate the population mean $({\mu})$ and population variance $({\sigma^2})$ To find the population mean , add up the values of all $6$ ages and divide by $6$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\mu} = \dfrac{11 + 3 + 9 + 8 + 2 + 15}{{6}} = {8\text{ years old}} $ Find the squared deviations from the mean for each sloth. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $11$ years $3$ years $9$ years $^2$ $3$ years $-5$ years $25$ years $^2$ $9$ years $1$ year $1$ year $^2$ $8$ years $0$ years $0$ years $^2$ $2$ years $-6$ years $36$ years $^2$ $15$ years $7$ years $49$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{9} + {25} + {1} + {0} + {36} + {49}} {{6}} $ $ {\sigma^2} = \dfrac{{120}}{{6}} = {20\text{ years}^2} $ The average sloth at the zoo is 8 years old. The population variance is 20 years $^2$.